Lorentzian Entanglement Growth Dynamics

• Lorentzian entanglement growth is the study of time-evolving quantum correlations in Lorentzian signature systems, encompassing holographic quenches, free field theories, and BTZ black hole analyses.
• It examines diverse observables such as spatial and time-like entropies, operator entanglement, and connected correlators to distinguish between entanglement build-up and propagation.
• The approach reveals distinct growth regimes—quadratic, linear, and logarithmic—and establishes rigorous speed limits and causal bounds in both holographic and many-body contexts.

Lorentzian entanglement growth denotes several related but non-identical problems concerning entanglement in real time or Lorentzian signature. In holographic quench dynamics it refers to the time evolution of entanglement entropy computed by covariant extremal surfaces in time-dependent asymptotically AdS spacetimes (Liu et al., 2013). In free scalar field theory it refers to the real-time growth of von Neumann and Rényi entropies after a global quench (Cotler et al., 2016). In a more specialized usage for the double Wick-rotated rotating BTZ black hole, it is defined by the coefficient of linear growth of time-like entanglement entropy, $\lambda_{\mathrm{TL}}=\frac{c}{6}r_+$ (Dai et al., 17 Apr 2026). The literature also treats operator-space entanglement under Heisenberg evolution, lightcone-modified correlation growth from initially entangled states, and several adjacent notions in which “Lorentzian” refers to a reservoir spectral density, a Lorentz transformation, or Lorentzian network histories rather than to a single canonical growth law (Correr et al., 26 Mar 2026, Kastner, 2015, Teklu, 2022, Metwally et al., 2014). This suggests that the subject is best organized by observable and physical setting rather than by terminology alone.

1. Scope, observables, and meanings of the term

The central ambiguity is that the literature does not attach the phrase to one invariant quantity. In some works the object is ordinary spatial entanglement entropy $S_{\mathcal A}(t)$ of a subregion; in others it is a time-like analogue, a Rényi entropy in Liouville space, or an equal-time connected correlator used as a proxy for information spreading. The distinction is substantive because the corresponding growth laws, bounds, and causal interpretations are different.

Usage in the literature	Quantity	Representative result
Global-quench entanglement	$S_{\mathcal A}(t)$, $S_q(t)$	Linear regime and geometry-dependent saturation (Cotler et al., 2016)
Holographic thermalization	$S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$	Quadratic, linear, memory-loss, and saturation regimes (Liu et al., 2013)
Time-like BTZ entropy	$S_{\mathrm{TL}}$	$\lambda_{\mathrm{TL}}=\frac{c}{6}r_+$ (Dai et al., 17 Apr 2026)
Local operator entanglement	$S_2[\rho^{(\mathcal O)}_{\mathscr A}(t)]$	Late-time volume law in chaotic systems (Correr et al., 26 Mar 2026)
Initially entangled correlation spreading	$\langle A(t)B(t)\rangle_{\mathrm c}$	LR-type bound with explicit initial-state term (Kastner, 2015)

A second distinction concerns whether the observable diagnoses entanglement build or entanglement move. In the many-body localized setting, the Wehrl-Rényi entropy is introduced precisely because it is invariant under qubit permutations/SWAPs and therefore isolates entanglement build from mere entanglement move (Xu et al., 20 May 2026). That separation does not appear in the usual spatial entanglement entropy of a subregion.

2. Real-time growth after quenches and the fate of lightcone structure

In free scalar field theory after a global quench, the subtracted entropy $$\hat S_{\mathcal A}(t)=S_{\mathcal A}(t)-S_{\mathcal A}(0)$$ is studied for strips in $S_{\mathcal A}(t)$0 spatial dimensions and for spheres in $S_{\mathcal A}(t)$1 spatial dimensions. In the scaling regime the leading growth takes the form

$S_{\mathcal A}(t)$2

with

$S_{\mathcal A}(t)$3

For spheres the leading extensive contribution saturates at $S_{\mathcal A}(t)$4, whereas for strips $S_{\mathcal A}(t)$5. At subleading order the numerics show an anomalous logarithmic growth $S_{\mathcal A}(t)$6 from the zero mode of the scalar (Cotler et al., 2016).

Long-range interacting spin systems modify the same real-time picture in a different way. In a transverse-field Ising chain with algebraically decaying couplings $S_{\mathcal A}(t)$7, the half-chain entropy grows linearly in time for relatively short-range interactions, roughly $S_{\mathcal A}(t)$8 and especially clearly for $S_{\mathcal A}(t)$9, but for most of the long-range regime $S_{\mathcal A}(t)$0 it grows only logarithmically, $S_{\mathcal A}(t)$1. Mutual information then shows the transition from delayed, front-like propagation to almost immediate short-time structure at long distance (Schachenmayer et al., 2013). A central lesson is that the breakdown of a strict light cone does not imply faster bipartite entanglement growth.

Initial entanglement changes the propagation problem even when the Hamiltonian is local. For arbitrary initial states in quantum lattice systems, the connected correlator obeys a bound of the form

$S_{\mathcal A}(t)$2

where $S_{\mathcal A}(t)$3 contains a Lieb-Robinson-type dynamical term and an explicit initial-state term $S_{\mathcal A}(t)$4. In this framework, pre-existing entanglement can make distant correlations appear much earlier than for product states, but the enhancement does not increase the Lieb-Robinson velocity itself (Kastner, 2015).

3. Holographic Lorentzian growth and speed limits

For homogeneous quenches in strongly coupled CFTs with gravity duals, the bulk geometry is AdS-Vaidya in the thin-shell limit, and entanglement entropy is computed by the covariant Hubeny-Rangamani-Takayanagi prescription. In the large-distance limit the extremal surface probes the geometry around and inside the event horizon, and the evolution exhibits four regimes: pre-local-equilibration quadratic growth, post-local-equilibration linear growth, a memory-loss regime, and a saturation regime (Liu et al., 2013). The early-time result is

$S_{\mathcal A}(t)$5

while for large regions after local equilibration

$S_{\mathcal A}(t)$6

In this setting the linear regime is controlled by critical extremal surfaces behind the horizon, and the slope is shape independent.

A later development turns these Lorentzian growth statements into rigorous speed limits in spatially uniform holographic states. In $S_{\mathcal A}(t)$7 CFT, for a region that is a union of $S_{\mathcal A}(t)$8 finite intervals,

$S_{\mathcal A}(t)$9

For higher dimensions, thin-shell planar-symmetric spacetimes give corresponding area-type and volume-type bounds for strips and balls, and for small subregions one obtains

$S_q(t)$0

The key structural result is the momentum-entanglement correspondence,

$S_q(t)$1

which identifies entanglement growth with bulk momentum crossing the HRT surface. The same analysis proves sharp bounds on the smallest radius an extremal surface can probe and shows that the tips of boundary-anchored extremal surfaces cannot lie in trapped regions (Folkestad et al., 2022).

4. Time-like entanglement entropy and the rotating BTZ definition

A distinct and narrower definition appears in the analysis of the double Wick rotation of the rotating BTZ black hole. There the field-theory object dual to the double Wick-rotated geometry is not an ordinary density matrix but a transition matrix with the usual shape at an imaginary chemical potential, and the geometric entropy is defined by tracing over a subsystem of that transition matrix (Dai et al., 17 Apr 2026). The crucial Lorentzian continuation exchanges space and time and leads to a time-like interval rather than a spatial interval on a constant-time slice.

The corresponding time-like entanglement entropy is

$S_q(t)$2

At late times,

$S_q(t)$3

and the coefficient

$S_q(t)$4

is what the paper calls the new Lorentzian entanglement growth (Dai et al., 17 Apr 2026). The comparison with the usual Lyapunov exponent,

$S_q(t)$5

is central: $S_q(t)$6 remains finite in the extremal limit where $S_q(t)$7. The construction is therefore not a quench calculation in the usual spatial-entanglement sense, but a Lorentzian, analytically continued, time-like entropy growth law tied to complexified modular flow.

5. Operator-space growth and structured initial states

Local Operator Entanglement extends Lorentzian entanglement growth into operator space. A traceless local Hermitian operator $S_q(t)$8 is evolved in the Heisenberg picture, vectorized into a doubled Hilbert space, and the $S_q(t)$9-Rényi entropy of the reduced operator-state is computed: $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$0 Chaotic short-range systems show LOE growth at most linearly in time, while integrable systems show only logarithmic growth. The main analytical result is not a derivation of the ramp but of the late-time saturation value: under a 4-non-resonance condition, the ETH ansatz, and a Haar-random replacement of Hamiltonian eigenstates in the final expression, the late-time LOE exhibits volume-law scaling (Correr et al., 26 Mar 2026).

Structured initial states in many-body localized systems provide a different generalization. In the random-field XXZ chain, a product state is first evolved under a chaotic Hamiltonian for a preparation time $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$1, and only then quenched to the target Hamiltonian. In the localized phase the hallmark growth law is

$S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$2

with half-chain saturation time of order $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$3. The net late-time growth of the half-chain second Rényi entropy is non-monotonic as a function of $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$4 for $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$5-polarized seeds: it first increases, then decreases, reflecting a crossover from LIOM magnetization relaxation to inter-site correlations. The same paper explicitly states that it does not provide a Lorentzian fitting function and does not claim Lorentzian time dependence of entanglement growth (Xu et al., 20 May 2026).

6. Adjacent usages, ambiguities, and non-equivalences

Several papers use nearby language in ways that should not be conflated with the real-time or time-like growth laws above. In continuous-variable open systems, “Lorentzian” refers to the reservoir spectral density, $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$6 or $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$7. The resulting non-Markovian dynamics can show entanglement sudden death, revivals, oscillatory backflow, and “always alive” behavior, but the mechanism is reservoir memory rather than a Lorentzian-signature entanglement law (Teklu, 2022).

In relativistic two-qubit models, Lorentz transformation changes the spin-state entanglement through Wigner rotation. The boost can degrade entanglement already present in the initial state and can also create nonzero spin entanglement from some states that are initially separable; partially entangled states are reported to be more robust than maximally entangled states (Metwally et al., 2014). Here “Lorentzian” refers to frame dependence under boosts, not to entanglement growth in many-body real time.

Other Lorentzian settings are again different in kind. Quantum fields confined to Lorentzian histories of freely falling networks recover an area law for vacuum entanglement, but the calculation is static and kinematic rather than a study of $S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)$8 after a quench (Giavoni et al., 2023). Scalar cosmological perturbations in Lorentzian group field theory are sourced by nearest-neighbor two-body entanglement, but the work does not compute entanglement entropy growth, mutual information spreading, or a time-dependent entanglement measure (Jercher et al., 2023). Lorentzian AdS wormholes clarify the structure of same-boundary and cross-boundary correlators and the distinction between coupling and entanglement, but they do not compute time-dependent entanglement entropy growth (Arias et al., 2010).

Taken together, these results establish that Lorentzian entanglement growth is not a single universal law. The literature contains at least three technically distinct cores: real-time growth of spatial entanglement after quenches, Lorentzian holographic speed limits for HRT surfaces, and the time-like entropy growth defined in the rotating BTZ context. Around those cores lies a wider zone of adjacent usages in which “Lorentzian” names a reservoir, a boost, or a causal background rather than the same dynamical observable.